reserve G for Group;
reserve A,B for non empty Subset of G;
reserve N,H,H1,H2 for Subgroup of G;
reserve x,a,b for Element of G;
reserve N1,N2 for Subgroup of G;

theorem
  for N be normal Subgroup of G holds N ` H1 * N ` H2 c= N ` (H1 * H2)
proof
  let N be normal Subgroup of G;
  let x be object;
  assume
A1: x in N ` H1 * N ` H2;
  then reconsider x as Element of G;
  consider x1, x2 be Element of G such that
A2: x = x1 * x2 & x1 in N ` H1 & x2 in N ` H2 by A1;
 x1 * N c= carr(H1) & x2 * N c= carr(H2) by A2,Th49;
  then (x1 * N) * (x2 * N) c= carr(H1) * carr(H2) by GROUP_3:4;
  then (x1 * x2) * N c= carr(H1) * carr(H2) by Th1;
  hence thesis by A2;
end;
